Generalized autocommuting probability of a finite group relative to its subgroups

Year 2020, Volume: 49 Issue: 1, 389 - 398, 06.02.2020

Parama Dutta Rajat Nath

https://doi.org/10.15672/hujms.568258

Abstract

Let $H \subseteq K$ be two subgroups of a finite group $G$  and $\mathrm{Aut}(K)$ the automorphism group of  $K$. In this paper, we consider the generalized autocommuting probability of $G$ relative to its subgroups $H$ and $K$, denoted by  ${Pr}_g(H,\mathrm{Aut}(K))$, which is the probability  that the autocommutator of a randomly chosen pair of elements, one from $H$ and the other from $\mathrm{Aut}(K)$, is equal to a given element $g \in K$. We study several properties as well as obtain several computing formulae of  this probability. As applications of the computing formulae, we also obtain several  bounds for ${Pr}_g(H,\mathrm{Aut}(K))$ and characterizations of some finite groups through ${Pr}_g(H,\mathrm{Aut}(K))$.

Keywords

automorphism group, autocommuting probability, autoisoclinism

References

• [1] H. Arora and R. Karan, What is the probability an automorphism fixes a group element?, Comm. Algebra, 45(3), 1141–1150, 2017.
• [2] A.K. Das and R.K. Nath, On generalized relative commutativity degree of a finite group, Int. Electron. J. Algebra, 7, 140–151, 2010.
• [3] P. Dutta and R.K. Nath, Autocommuting probabilty of a finite group, Comm. Algebra, 46 (3), 961–969, 2018.
• [4] P. Dutta and R.K. Nath, On generalized autocommutativity degree of finite groups, Hacet. J. Math. Stat. 48 (2), 472–478, 2019.
• [5] P. Hall, The classification of prime power groups, J. Reine Angew. Math. 182, 130– 141, 1940.
• [6] P.V. Hegarty, The absolute centre of a group, J. Algebra, 169 (3), 929–935, 1994.
• [7] C.J. Hillar and D.L. Rhea, Automorphism of finite abelian groups, Amer. Math. Monthly, 114 (10), 917–923, 2007.
• [8] M.R.R. Moghaddam, M.J. Sadeghifard and M. Eshrati, Some properties of autoisoclinism of groups, Fifth International group theory conference, Islamic Azad University, Mashhad, Iran, 13-15 March 2013.
• [9] M.R.R. Moghaddam, F. Saeedi and E. Khamseh, The probability of an automorphism fixing a subgroup element of a finite group, Asian-Eur. J. Math. 4 (2), 301–308, 2011.
• [10] R.K. Nath and A.K. Das, On a lower bound of commutativity degree, Rend. Circ. Mat. Palermo, 59 (1), 137–142, 2010.
• [11] R.K. Nath and M.K. Yadav, Some results on relative commutativity degree, Rend. Circ. Mat. Palermo, 64 (2), 229–239, 2015.
• [12] M.R. Rismanchian and Z. Sepehrizadeh, Autoisoclinism classes and autocommutativity degrees of finite groups, Hacet. J. Math. Stat. 44 (4), 893–899, 2015.
• [13] G.J. Sherman, What is the probability an automorphism fixes a group element?, Amer. Math. Monthly, 82, 261–264, 1975.